I understand that current through the base electrode will trigger a transistor to act like a switch. Is there a way to calculate what is the minimum current (or voltage?) that is required to trigger that switch?

On the datasheet I see a "base-emitter saturation voltage" with 0.6V under "min". Is that it?

If you want the transistor to be fully on (saturated) you must ensure that the current flowing into the base is at least the maximum collector current divided by the typical current gain (Ic / hfe). You must also be careful to limit the collector current or else you will damage the transistor.

The 0.6V is the voltage that will be present across the base emitter junction when the transistor is biased on.